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Section 1.2 Applications of Linear Equations: Modeling

Before Starting this Section, Review

1 Distributive property (Section P.4 , page 41)
2 Arithmetic of fractions, least common denominator (Section P.5 , page 54)
3 Linear Equations (Section 1.1 , page 83)

Objectives

1 Learn modeling procedures for solving applied problems.
2 Use linear equations to model applied problems.

Bomb Threat on the Queen Elizabeth 2

While travelling from New York to Southampton, England, the captain of the Queen Elizabeth 2 received a message that a bomb was on board and that it was timed to go off during the voyage. A search by crew members proved fruitless, so members of the Royal Marine Special Boat Squadrons bomb disposal unit were flown out on a Royal Air Force Hercules aircraft and parachuted into the Atlantic near the ship, which was 1000 miles from Britain, traveling toward Britain at 32 miles per hour. If the Hercules aircraft could average 300 miles per hour, how long would the passengers have to wait for help to arrive? We learn the answer in Example 6, using the methods of this section.

Solving Applied Problems

1 Learn modeling procedures for solving applied problems.

Practical problems are often introduced through verbal descriptions. Frequently, they describe a situation, perhaps in a physical or financial setting, and ask for a value for some specific quantity that will favorably resolve the situation. The process of translating a practical problem into a mathematical one is called mathematical modeling. Generally, we use variables to represent the quantities identified in the verbal description and represent relationships between these quantities with algebraic expressions or equations. Ideally, we reduce the problem to an equation that, when solved, also solves the practical problem. Then we solve the equation and check our result against the practical situation presented in the problem.

Here is a general strategy for solving applied problems.

Procedure for Solving Applied Problems

Step 1 Read the problem as many times as needed to understand it thoroughly. Pay close attention to the question asked to help you identify the quantity the variable should represent.
Step 2 Assign a variable to represent the quantity you are seeking and, when necessary, express all other unknown quantities in terms of this variable. Frequently, it is helpful to draw a diagram to illustrate the problem or to set up a table to organize the information.
Step 3 Write an equation (the model) that describes the situation. Look for a relationship between the unknown quantities in Step 2.
Step 4 Solve the equation.
Step 5 Answer the question asked in the problem.
Step 6 Check the answer against the description of the original problem (not just the equation solved in Step 4).

Example 1 Modeling a Pre-sale Price

A sports complex is offering a fall semester fitness program to all college freshmen at a 20% discount. If the discounted price is $144, what was the original price?

Solution

Step 2 $\begin{array}{l} Let x = the original price . & This is the amount you want to find . \end{array}$ $\begin{array}{l} Let x = the original price . & This is the amount you want to find . \end{array}$

Then the discounted price is the original price minus 20%, or $x - 0.2 x .$ $x - 0.2 x .$
Step 3 $\begin{matrix} So x - 0.2 x = 144 & Discounted price = $ 144 \end{matrix}$ $\begin{matrix} So x - 0.2 x = 144 & Discounted price = $ 144 \end{matrix}$
Step 4 $\begin{array}{rcll} 0.8 x & = & 144 & x - 0.2 x = (1 - 0.2) x = 0.8 x \\ x & = & 180 & Divide both sides by 0.8, \frac{144}{0.8} = 180 . \end{array}$ $\begin{array}{rcll} 0.8 x & = & 144 & x - 0.2 x = (1 - 0.2) x = 0.8 x \\ x & = & 180 & Divide both sides by 0.8, \frac{144}{0.8} = 180 . \end{array}$
Step 5 The original price of the fitness program was $180.
Step 6 Check: If the original price was $180, since 20% of $180 is $36, we have $180 minus $36, or $144.

Practice Problem 1

A bookstore is selling a backpack at a 15% discount. If the discounted price is $29.75, what was the original price?

Geometry

2 Use linear equations to model applied problems.

Common geometric figures and formulas are given on the final pages of the text.

Example 2 Modeling a Geometry Problem

The length of a rectangular soccer field is 30 yards less than twice its width. Find the dimensions of the field assuming that its perimeter is 360 yards.

Solution

Step 2 $\begin{array}{rcl} Let x & = & the width of the field . \\ Then 2 x - 30 & = & the length of the field . (See the following figure .) \end{array}$ $\begin{array}{rcl} Let x & = & the width of the field . \\ Then 2 x - 30 & = & the length of the field . (See the following figure .) \end{array}$
Step 3 $\begin{array}{l} \begin{array}{rcl} Perimeter & = & sum of the lengths of the four sides of the rectangle \\ = & (2 x - 30) + x + (2 x - 30) + x \end{array} \\ \begin{array}{rcll} Perimeter & = & 360 yards & Given \\ (2 x - 30) + x + (2 x - 30) + x & = & 360 & \begin{array}{l} Replace the verbal description of \\ perimeter with the algebraic \\ expression . \end{array} \end{array} \end{array}$ $\begin{array}{l} \begin{array}{rcl} Perimeter & = & sum of the lengths of the four sides of the rectangle \\ = & (2 x - 30) + x + (2 x - 30) + x \end{array} \\ \begin{array}{rcll} Perimeter & = & 360 yards & Given \\ (2 x - 30) + x + (2 x - 30) + x & = & 360 & \begin{array}{l} Replace the verbal description of \\ perimeter with the algebraic \\ expression . \end{array} \end{array} \end{array}$
Step 4

$\begin{array}{rcll} 6 x - 60 & = & 360 & Combine terms . \\ 6 x & = & 420 & Add 60 to both sides and simplify . \\ x & = & \frac{420}{6} = 70 & Divide each side by 6 . \end{array}$ $\begin{array}{rcll} 6 x - 60 & = & 360 & Combine terms . \\ 6 x & = & 420 & Add 60 to both sides and simplify . \\ x & = & \frac{420}{6} = 70 & Divide each side by 6 . \end{array}$
Step 5 $\begin{array}{l} The width of the field = 70 yards, and \\ the length of the field = 2 (70) - 30 = 140 - 30 = 110 yards . \end{array}$ $\begin{array}{l} The width of the field = 70 yards, and \\ the length of the field = 2 (70) - 30 = 140 - 30 = 110 yards . \end{array}$

The dimensions of the soccer field are 110 by 70 yards.
Step 6 Check: The perimeter is $110 + 70 + 110 + 70 = 360$ $110 + 70 + 110 + 70 = 360$ yards.

Practice Problem 2

The length of a rectangle is 5 m more than twice its width. Find the dimensions of the rectangle, assuming that its perimeter is 28 m.

Finance

Example 3 Modeling an Investment Problem

Tyrick invests $15,000, some in stocks and the rest in bonds. If he invests twice as much in stocks as he does in bonds, how much does he invest in each?

Solution

Step 2 Let $x = the amount invested in stocks .$ $x = the amount invested in stocks .$ The rest of the $15,000 investment $(15,000 - x)$ $(15,000 - x)$ is invested in bonds.
Step 3 We have one more important piece of information to use, as follows:

$\begin{array}{rcll} Amount invested in stocks, x = \begin{array}{l} Twice the amount invested \\ in bonds, 2 (15,000 - x) \end{array} \\ \begin{array}{rcll} x & = & 2 (15,000 - x) & Replace the verbal descriptions \\ with algebraic expressions . \end{array} \end{array}$ $\begin{array}{rcll} Amount invested in stocks, x = \begin{array}{l} Twice the amount invested \\ in bonds, 2 (15,000 - x) \end{array} \\ \begin{array}{rcll} x & = & 2 (15,000 - x) & Replace the verbal descriptions \\ with algebraic expressions . \end{array} \end{array}$
Step 4

$\begin{array}{rcll} x & = & 30,000 - 2 x & Distributive property \\ 3 x & = & 30,000 & Add 2 x to both sides . \\ x & = & 10,000 & Divide both sides by 3 . \end{array}$ $\begin{array}{rcll} x & = & 30,000 - 2 x & Distributive property \\ 3 x & = & 30,000 & Add 2 x to both sides . \\ x & = & 10,000 & Divide both sides by 3 . \end{array}$
Step 5 Tyrick invests $10,000 in stocks and $$ 15,000 - $ 10,000 = $ 5000$ $$ 15,000 - $ 10,000 = $ 5000$ in bonds.
Step 6 Check in the original problem. If Tyrick invests $10,000 in stocks and $5000 in bonds, then because $2 ($ 5000) = $ 10,000,$ $2 ($ 5000) = $ 10,000,$ he invested twice as much in stocks as he did in bonds.

Practice Problem 3

Repeat Example 3 , but Tyrick now invests three times as much in stocks as he does in bonds.

Interest is money paid for the use of borrowed money. For example, while your money is on deposit at a bank, the bank pays you for the right to use the money in your account. The money you are paid by the bank is interest on your deposit. On any loan, the money borrowed for use (and on which interest is paid) is called the principal. The interest rate is the percent of the principal charged for its use for a specific time period. The most straightforward type of interest computation is described next.

Example 4 Modeling a Simple Interest Problem

Ms. Sharpy invests a total of $10,000 in blue-chip and technology stocks. At the end of a year, the blue chips returned 12% and the technology stocks returned 8% on the original investments. How much was invested in each type of stock if the total interest earned was $1060?

Solution

Step 1 We are asked to find two amounts: that invested in blue-chip stocks and that invested in technology stocks. If we know how much was invested in blue-chip stocks, then we know that the rest of the $10,000 was invested in technology stocks.
Step 2 Let $x = amount$ $x = amount$ invested in blue-chip stocks. Then

$10,000 - x = amount invested in technology stocks .$ $10,000 - x = amount invested in technology stocks .$

Step 3 We organize our information in the following table.

Investment	Principal `P`	Rate `r`	Time `t`	Interest $I = P r t$ $I = P r t$
Blue Chip	`x`	0.12	1	$0.12 x$ $0.12 x$
Technology	$10,000 - x$ $10,000 - x$	0.08	1	$0.08 (10,000 - x)$ $0.08 (10,000 - x)$

Interest from blue chip + Interest from technology = Total Interest

$Interest from blue chip + Interest from technology = Total Interest$

Step 4

$\begin{array}{rcll} 0.12 x + 0.08 (10,000 - x) & = & 1060 & Replace descriptions with \\ algebraic expressions . \\ 12 x + 8 (10,000 - x) & = & 106,000 & Multiply by 100 to \\ eliminate decimals . \\ 12 x + 80,000 - 8 x & = & 106,000 & Distributive property \\ 4 x & = & 26,000 & Combine like terms; \\ ` & subtract 80,000 from \\ both sides . \\ x & = & 6500 & Divide by 4 to get the \\ amount in blue-chip \\ stocks . \\ 10,000 - x & = & 10,000 - 6500 & Replace x with 6500 . \\ = & 3500 & Substitute to get the \\ amount in technology \\ stocks . \end{array}$ $\begin{array}{rcll} 0.12 x + 0.08 (10,000 - x) & = & 1060 & Replace descriptions with \\ algebraic expressions . \\ 12 x + 8 (10,000 - x) & = & 106,000 & Multiply by 100 to \\ eliminate decimals . \\ 12 x + 80,000 - 8 x & = & 106,000 & Distributive property \\ 4 x & = & 26,000 & Combine like terms; \\ ` & subtract 80,000 from \\ both sides . \\ x & = & 6500 & Divide by 4 to get the \\ amount in blue-chip \\ stocks . \\ 10,000 - x & = & 10,000 - 6500 & Replace x with 6500 . \\ = & 3500 & Substitute to get the \\ amount in technology \\ stocks . \end{array}$
Step 5 Ms. Sharpy invests $3500 in technology stocks and $6500 in blue-chip stocks.
Step 6 Check in the original problem.

$\begin{array}{rcl} 12 % of $ 6500 is 0.12 (6500) & = & $ 780 \\ and 8 % of $ 3500 is 0.08 (3500) & = & $ 280 . \end{array}$ $\begin{array}{rcl} 12 % of $ 6500 is 0.12 (6500) & = & $ 780 \\ and 8 % of $ 3500 is 0.08 (3500) & = & $ 280 . \end{array}$

Thus, the total interest earned is $$780 + $280 = $1060$ $$780 + $280 = $1060$ .

Practice Problem 4

One-fifth of my capital is invested at 5%, one-sixth of my capital at 8%, and the rest at 10% per year. If the annual interest on my capital is $130, what is my capital?

Uniform Motion

If you drive 60 miles in two hours, your average speed is $\frac{60}{2} = 30$ $\frac{60}{2} = 30$ miles per hour. If you multiply your average speed for the trip (30 miles per hour) by the time elapsed (two hours), you get the distance driven, 60 miles $(60 = 2 \cdot 30)$ $(60 = 2 \cdot 30)$ . When “rate” refers to the average speed of an object, use the following formula.

Warning

Care must be taken to make sure that units of measurement are consistent. If, for example, the rate is given in miles per hour, then the time should be given in hours. If the rate is given in miles per hour and the time is given as 15 minutes, change 15 minutes to $\frac{1}{4}$ $\frac{1}{4}$ hour before using the uniform-motion formula.

Example 5 Modeling a Uniform-Motion Problem

A motorcycle police officer is chasing a car that is speeding at 70 miles per hour. The police officer is 1 mile behind the car and is traveling 80 miles per hour. How long will it be before the officer overtakes the car?

Solution

Step 2 We are asked to find the amount of time before the officer overtakes the car. Draw a sketch to help visualize the problem. See Figure 1.1 .

Let $t = the time it takes the officer to overtake the car .$ $t = the time it takes the officer to overtake the car .$ Organize the information in a table.

Object	`r` (miles per hour)	`t` (hours)	$d = r t$ $d = r t$ (miles)
Car	70	`t`	70`t`
Motorcycle	80	`t`	80`t`

Step 3 Because the distance the motorcycle travels is 1 mile more than the car travels, we have

$\begin{array}{rcll} 80 t & = & 70 t + 1 & The motorcycle ’ s distance is 1 mile more than the \\ car ’ s distance . \end{array}$ $\begin{array}{rcll} 80 t & = & 70 t + 1 & The motorcycle ’ s distance is 1 mile more than the \\ car ’ s distance . \end{array}$
Step 4

$\begin{array}{rcll} 80 t - 70 t & = & 1 & Subtract 70 t from both sides . \\ 10 t & = & 1 & Simplify . \\ t & = & \frac{1}{10} & Divide both sides by 10 . \end{array}$ $\begin{array}{rcll} 80 t - 70 t & = & 1 & Subtract 70 t from both sides . \\ 10 t & = & 1 & Simplify . \\ t & = & \frac{1}{10} & Divide both sides by 10 . \end{array}$
Step 5 The time required to overtake the car is

$t = \frac{1}{10} hour = 0.1 hour (or 6 minutes)$ $t = \frac{1}{10} hour = 0.1 hour (or 6 minutes)$
Step 6 Check in the original problem. If the police officer travels for 0.1 of an hour at 80 miles per hour, he travels $d = (80) (0.1) = 8$ $d = (80) (0.1) = 8$ miles. The car traveling for this same 0.1 of an hour at 70 miles per hour travels $(70) (0.1) = 7$ $(70) (0.1) = 7$ miles. Because the officer travels 1 more mile during that time than the car does (the motorcycle started 1 mile behind the car), the officer does overtake the car.

Practice Problem 5

A train 130 meters long crosses a bridge in 21 seconds. The train is traveling at 25 meters per second. What is the length of the bridge?

Example 6 Dealing with a Bomb Threat on the Queen Elizabeth II

In the introduction to this section, the Queen Elizabeth II was 1000 miles from Britain, traveling toward Britain at 32 miles per hour. A Hercules aircraft was flying from Britain directly toward the ship, averaging 300 miles per hour. How long would the passengers have to wait for the aircraft to meet the ship?

Solution

Step 1 The initial separation between the ship and the aircraft is 1000 miles.
Step 2 Let $t = time$ $t = time$ elapsed when ship and aircraft meet.

Then $32 t = distance the ship traveled$ $32 t = distance the ship traveled$ and $300 t = distance the aircraft$ $300 t = distance the aircraft$ traveled.
Step 3

$\begin{array}{c} Distance the ship traveled + Distance the aircraft traveled = 1000 miles \\ \begin{array}{rcll} 32 t + 300 t & = & 1000 & Replace descriptions with algebraic \\ expressions . \end{array} \end{array}$ $\begin{array}{c} Distance the ship traveled + Distance the aircraft traveled = 1000 miles \\ \begin{array}{rcll} 32 t + 300 t & = & 1000 & Replace descriptions with algebraic \\ expressions . \end{array} \end{array}$
Step 4

$\begin{array}{rcll} 332 t & = & 1000 & Combine like terms . \\ t & = & \frac{1000}{332} \approx 3.01 & Divide both sides by 332 . \end{array}$ $\begin{array}{rcll} 332 t & = & 1000 & Combine like terms . \\ t & = & \frac{1000}{332} \approx 3.01 & Divide both sides by 332 . \end{array}$
Step 5 The aircraft and ship meet after about three hours.

Practice Problem 6

How long would it take for the aircraft and ship in Example 6 to meet if the aircraft averaged 350 miles per hour and was 955 miles away?

Work Rate

Work-rate problems use an idea much like that of uniform motion. In problems involving rates of work, we assume that the work is done at a uniform rate.

Work Rate

The portion of a job completed per unit of time is called the rate of work.

If a job can be completed in x units of time (seconds, hours, days, and so on), then the portion of the job completed in one unit of time (1 second, 1 hour, 1 day, and so on) is $\frac{1}{x}$ $\frac{1}{x}$ . The portion of the job completed in t units of time (t seconds, t hours, t days, and so on) is $t \cdot \frac{1}{x}$ $t \cdot \frac{1}{x}$ . When the portion of the job completed is 1, the job is done.

Example 7 Modeling a Work-Rate Problem

One copy machine copies twice as fast as another. If both copiers work together, they can finish a particular job in two hours. How long would it take each copier, working alone, to do the job?

Solution

Step 1 Because the speed of one copier is twice the speed of the other, if we find how long it takes the faster copier to do the job working alone, the slower copier will take twice as long.

Step 2 $\begin{array}{rcl} Let x & = & number of hours for the faster copier to complete the job alone . \\ 2 x & = & number of hours for the slower copier to complete the job alone . \\ \frac{1}{x} & = & portion of the job the faster copier does in one hour . \\ \frac{1}{2 x} & = & portion of the job the slower copier does in one hour . \end{array}$ $\begin{array}{rcl} Let x & = & number of hours for the faster copier to complete the job alone . \\ 2 x & = & number of hours for the slower copier to complete the job alone . \\ \frac{1}{x} & = & portion of the job the faster copier does in one hour . \\ \frac{1}{2 x} & = & portion of the job the slower copier does in one hour . \end{array}$

The time it takes both copiers, working together, to do the job is two hours. Multiplying the portion of the job each copier does in one hour by 2 will give the portion of the completed job done by each copier. We organize our information in a table.

	Portion Done in One Hour	Time to Complete Job Working Together	Portion Done by Each Copier
Faster copier	$\frac{1}{x}$ $\frac{1}{x}$	2	$2 (\frac{1}{x}) = \frac{2}{x}$ $2 (\frac{1}{x}) = \frac{2}{x}$
Slower copier	$\frac{1}{2 x}$ $\frac{1}{2 x}$	2	$2 (\frac{1}{2 x}) = \frac{1}{x}$ $2 (\frac{1}{2 x}) = \frac{1}{x}$

Because the job is completed in two hours, the sum of the portion of the job completed by the faster copier in two hours and the portion of the job completed by the slower copier in two hours is 1. The model equation can now be written and solved.

Step 3 $\begin{matrix} \frac{2}{x} + \frac{1}{x} & = & 1 & Faster copier portion + slower copier portion = 1 . \end{matrix}$ $\begin{matrix} \frac{2}{x} + \frac{1}{x} & = & 1 & Faster copier portion + slower copier portion = 1 . \end{matrix}$
Step 4 $\begin{array}{rcll} 2 + 1 & = & x & Multiply both sides by x . \\ 3 & = & x \end{array}$ $\begin{array}{rcll} 2 + 1 & = & x & Multiply both sides by x . \\ 3 & = & x \end{array}$
Step 5 The faster copier could complete the job, working alone, in three hours. The slower copier would require $2 (3) = 6$ $2 (3) = 6$ hours to complete the job.
Step 6 Check in the original problem. The faster copier completes $\frac{1}{3}$ $\frac{1}{3}$ of the job in one hour. Similarly, the slower copier completes $\frac{1}{6}$ $\frac{1}{6}$ of the job in one hour.

So, in two hours, the faster copier has completed $2 (\frac{1}{3}) = \frac{2}{3}$ $2 (\frac{1}{3}) = \frac{2}{3}$ of the job and the slower copier has completed $2 (\frac{1}{6}) = \frac{2}{6} = \frac{1}{3}$ $2 (\frac{1}{6}) = \frac{2}{6} = \frac{1}{3}$ of the job. Because $\frac{2}{3} + \frac{1}{3} = 1$ $\frac{2}{3} + \frac{1}{3} = 1$ , the copiers have completed the job in two hours.

Practice Problem 7

A couple took turns washing their sports car on weekends. Jim could usually wash the car in 45 minutes, whereas Anita took 30 minutes to do the same job. One weekend they were in a hurry to go to a party, so they worked together. How long did it take them?

Mixtures

Mixture problems require combining various ingredients, such as water and antifreeze, to produce a blend with specific characteristics.

Example 8 Modeling a Mixture Problem

A full 6-quart radiator contains 75% water and 25% pure antifreeze. How much of this mixture should be drained and replaced by pure antifreeze so that the resulting 6-quart mixture is 50% pure antifreeze?

Solution

Step 1 We are asked to find the quantity of the radiator mixture that should be drained and replaced by pure antifreeze. The mixture we drain is only 25% pure antifreeze, but it is replaced with 100% pure antifreeze.
Step 2 Because we want to find the quantity of mixture drained, let

$x = number of quarts of mixture drained .$ $x = number of quarts of mixture drained .$

$\begin{array}{rcl} Then x & = & number of quarts of pure antifreeze added and \\ 0.25 x & = & number of quarts of pure antifreeze drained . \end{array}$ $\begin{array}{rcl} Then x & = & number of quarts of pure antifreeze added and \\ 0.25 x & = & number of quarts of pure antifreeze drained . \end{array}$

Now we track the pure antifreeze.

$\begin{array}{c} \begin{array}{l} Pure antifreeze \\ in final mixture \end{array} & = & \begin{array}{l} Pure antifreeze in \\ original mixture \end{array} & - & \begin{array}{l} Pure antifreeze \\ drained from \\ original mixture \end{array} & + & \begin{array}{l} Pure \\ antifreeze \\ added \end{array} \\ (50 % of 6) & = & (25 % of 6) & - & (25 % of x) & + & x \end{array}$ $\begin{array}{c} \begin{array}{l} Pure antifreeze \\ in final mixture \end{array} & = & \begin{array}{l} Pure antifreeze in \\ original mixture \end{array} & - & \begin{array}{l} Pure antifreeze \\ drained from \\ original mixture \end{array} & + & \begin{array}{l} Pure \\ antifreeze \\ added \end{array} \\ (50 % of 6) & = & (25 % of 6) & - & (25 % of x) & + & x \end{array}$
Step 3 $\begin{array}{rcll} (0.5) (6) & = & (0.25) (6) - 0.25 x + x & Replace descriptions with algebraic \\ expressions . \end{array}$ $\begin{array}{rcll} (0.5) (6) & = & (0.25) (6) - 0.25 x + x & Replace descriptions with algebraic \\ expressions . \end{array}$
Step 4 $\begin{array}{rcll} 3 & = & 1.5 + 0.75 x & Collect like terms; combine constants . \\ 1.5 & = & 0.75 x & Subtract 1.5 from both sides . \\ 2 & = & x & Divide both sides by 0.75 . \end{array}$ $\begin{array}{rcll} 3 & = & 1.5 + 0.75 x & Collect like terms; combine constants . \\ 1.5 & = & 0.75 x & Subtract 1.5 from both sides . \\ 2 & = & x & Divide both sides by 0.75 . \end{array}$
Step 5 Drain 2 quarts of mixture from the radiator.
Step 6 Check in the original problem. Draining 2 quarts from the original 6 quarts leaves 4 quarts of mixture in the radiator. This mixture consists of 1 quart of pure antifreeze (25%) and 3 quarts of water (75%). Adding 2 quarts of antifreeze to the radiator produces 3 quarts of antifreeze mixed with 3 quarts of water. This is the desired 50% pure antifreeze solution.

Practice Problem 8

How many gallons of 40% sulfuric acid solution should be mixed with 20% sulfuric acid solution to obtain 50 gallons of 25% sulfuric acid solution?

Section 1.2 Exercises

Concepts and Vocabulary

If the sides of a rectangle are a and b units, the perimeter of the rectangle is units.
The formula for simple interest is $I = P r t,$ $I = P r t,$ where $P = \underline{}, r = \underline{},$ $P = \underline{}, r = \underline{},$ and $t = \underline{} .$ $t = \underline{} .$
The distance d traveled by an object moving at rate r for time t is given by $d = \underline{} .$ $d = \underline{} .$
The portion of a job completed per unit of time is called the .
True or False. After a 30% discount the price of a bike is $238. The original price can be found by solving the equation $0.3 x = 238 .$ $0.3 x = 238 .$
True or False. If your first two test scores are 77 and 81, to find the score needed on a third test to achieve an average of 80, you can solve the equation

$\frac{77 + 81 + x}{3} = 80 .$ $\frac{77 + 81 + x}{3} = 80 .$
True or False. If $100 is invested at 5% for three years, the interest $I = (100) (5) (3) dollars$ $I = (100) (5) (3) dollars$ .
True or False. If an object is traveling at a uniform rate of 60 ft/sec, the distance covered in 15 minutes is $d = (60) (15) ft$ $d = (60) (15) ft$ .

Building Skills

In Exercises 9–18, write an algebraic expression for the specified quantity.

Desmond buys in-line skates for $327.62, including sales tax of $6 \frac{1}{2} %$ $6 \frac{1}{2} %$ . Let $x = the price$ $x = the price$ of in-line skates before tax. Write an algebraic expression in x for “the tax paid on the in-line skates.”
A lamp manufacturing company produces 1600 reading lamps a day when it operates in two shifts. The first shift produces only 5/6 as many lamps as the second shift. Let $x = the number$ $x = the number$ of lamps produced per day by the second shift. Write an algebraic expression in x for the number of lamps produced per day by the first shift.
Kim invests $22,000, some in stocks and the rest in bonds. Let $x = amount invested$ $x = amount invested$ in stocks. Write an algebraic expression in x for “the amount invested in bonds.”
An air conditioning repair bill for $229.50 showed a charge of $72.00 for parts, with the remainder of the charge for labor. Let $x = number of hours$ $x = number of hours$ of labor it took to repair the air conditioner. Write an algebraic expression in x for the labor charge per hour.
Natasha can pick an eight-tree section of orange trees in four hours working alone. It takes her brother five hours to pick the same section working alone.

Let $t = amount of time$ $t = amount of time$ it takes Natasha and her brother to pick the eight-tree section working together. Write an algebraic expression in t for
1. the portion of the job completed by Natasha in t hours.
2. the portion of the job completed by Natasha’s brother in t hours.
Jermaine can process a batch of tax forms in three hours working alone. Ralph can process a batch of tax forms in two hours working alone.

Let $t = amount of time$ $t = amount of time$ it takes Jermaine and Ralph to process a batch of tax forms working together. Write an algebraic expression in t for
1. the portion of the batch of tax forms completed by Jermaine in t hours.
2. the portion of the batch of tax forms completed by Ralph in t hours.
Walnuts sell for $7.40 per pound, and raisins sell for $4.60 per pound. Let $x = number of pounds$ $x = number of pounds$ of raisins in a 10-pound mixture of walnuts and raisins. Write an algebraic expression in x for
1. the number of pounds of walnuts in the 10-pound mixture.
2. the value of the raisins in the 10-pound mixture.
3. the value of the walnuts in the 10-pound mixture.
Dried pears sell for $5.50 per pound, and dried apricots sell for $6.25 per pound. Let $x = number of pounds$ $x = number of pounds$ of dried pears in a 15-pound mixture of dried pears and dried apricots. Write an algebraic expression in x for
1. the number of pounds of dried apricots in the 15-pound mixture.
2. the value of the dried apricots in the 15-pound mixture.
3. the value of the dried pears in the 15-pound mixture.
A pharmacist has 8 liters of a mixture that is 10% alcohol. To strengthen the mixture, the pharmacist adds pure alcohol.

Let $x = number of liters$ $x = number of liters$ of pure alcohol added to the 8-liter mixture. Write an algebraic expression in x for
1. the number of liters of mixture the pharmacist has after the pure alcohol is added.
2. the number of liters of alcohol in the mixture obtained by adding the pure alcohol to the original 8-liter mixture.
A gallon of a 25% saline solution is to be diluted by adding distilled water to it. Let $x = number of gallons$ $x = number of gallons$ of distilled water added to the gallon of 25% saline solution. Write an algebraic expression in x for
1. the number of gallons of saline solution after the distilled water is added.
2. the percent saline solution obtained by adding the distilled to the original 25% solution.

Applying the Concepts

In Exercises 19–70, use mathematical modeling techniques to solve the problems.

Pre-sale pricing. A computer is on sale for $1192. If the original price was reduced by 20%, what was the original price of the computer?
Pre-sale pricing. A pair of sneakers is on sale for $68. If the original price was reduced by 15%, what was the original price of the sneakers?
Test scores. Kylee’s first three test scores are 81, 75, and 77. What must Kylee score on the next test to raise the average to 80?
Test scores. Jocelyn’s first three test scores are 85, 76, and 88. If the final exam counts as two tests, what must Jocelyn score on the final exam to raise her average to 85?
Geometry. The perimeter of a rectangle is 80 ft. Find its dimensions, assuming that its length is 5 ft less than twice its width.
Geometry. The width of a rectangle is 3 ft more than one-half its length, and the perimeter of the rectangle is 36 ft. Find its dimensions.
Investment. If P dollars are invested at a simple interest rate r (in decimals), the amount A that will be available after t years is

$A = P + P r t .$ $A = P + P r t .$

If $500 is invested at a rate of 6%, how long will it be before the amount of money available is $920?

[Hint: Use $r = 0.06 .$ $r = 0.06 .$ ]
Investment. Using the formula from Exercise 25, determine the amount of money that was invested, assuming that $2482 resulted from a 10-year investment at 7%.
Estimating a mortgage note. A rule of thumb for estimating the maximum affordable monthly mortgage note M for prospective homeowners is

$M = \frac{m - b}{4},$ $M = \frac{m - b}{4},$

where m is the gross monthly income and b is the total of the monthly bills. What must the gross monthly income be to justify a monthly note of $427 if monthly bills are $302?
Engineering degrees. Among a group of engineers who had received master’s degrees, the annual salary S in dollars was related to two numbers b and a, where

$\begin{array}{rcl} b & = & the number of years of work experience before \\ receiving the degree and \\ a & = & the number of years of work experience after \\ receiving the degree . \end{array}$ $\begin{array}{rcl} b & = & the number of years of work experience before \\ receiving the degree and \\ a & = & the number of years of work experience after \\ receiving the degree . \end{array}$

The relationship between S, a, and b is given by

$S = 48,917 + 1080 b + 1604 a .$ $S = 48,917 + 1080 b + 1604 a .$

If an engineer had worked eight years before earning her master’s degree and was earning $67,181, how many years did she work after receiving her degree?
Digital cameras. The price P (in dollars) for which a manufacturer will sell a digital camera is related to the number q of cameras ordered by the formula

$P = 200 - 0.02 q, for 100 \leq q \leq 2000 .$ $P = 200 - 0.02 q, for 100 \leq q \leq 2000 .$

How many cameras must be ordered for a retailer to pay $170 per camera?
Boyle’s Law. If a volume $V_{1}$ $V_{1}$ of a dry gas under pressure $P_{1}$ $P_{1}$ is subjected to a new pressure $P_{2},$ $P_{2},$ the new volume $V_{2}$ $V_{2}$ of the gas is given by

$V_{2} = \frac{V_{1} P_{1}}{P_{2}} .$ $V_{2} = \frac{V_{1} P_{1}}{P_{2}} .$

Assuming that $V_{2}$ $V_{2}$ is 200 cubic centimeters, $V_{1}$ $V_{1}$ is 600 cubic centimeters, and $P_{1}$ $P_{1}$ is 400 millimeters of mercury, find $P_{2} .$ $P_{2} .$
Chain store properties. A fast-food chain bought two pieces of land for new stores. The more expensive piece of land costs $23,000 more than the less expensive piece of land, and the two pieces together cost $147,000. How much did each piece cost?
Administrative salaries. The manager at a wholesale outlet earns $450 more per month than the assistant manager. If their combined salary is $3700 per month, what is the monthly salary of each?
Lottery ticket sales. The lottery ticket sales at Quick Mart for August were 10% above the sales for July. If Quick Mart sold a total of 1113 lottery tickets during July and August, how many tickets were sold each month?
Sales commission. Jan’s commission for February was $15 more than half her commission for March, and her total commission for the two months was $633. Find her commission for each month.
Inheritance. An estate valued at $225,000 is to be divided between two sons so that the older son receives four times as much as the younger son. Find each son’s share of the estate.
TV game show. Kevin won $735,000 on a TV game show. He decided to keep a certain amount for himself, give one-half the amount he kept for himself to his daughter, and give one-fourth the amount he kept for himself to his dad. How much did each person get?
Grades. Your grades in the three tests in College Algebra are 87, 59, and 73.
1. How many points do you need on the final to average 75?
2. Find part (a), assuming that the final carries double weight.
Real estate investment. A real estate agent invested a total of $4200. Part was invested in a high-risk real-estate venture, and the rest was invested in a savings and loan. At the end of a year, the high-risk venture returned 15% on the investment and the savings and loan returned 8% on the investment. Find the amount invested in each assuming that the agent’s total income from the investments was $448.
Tax shelter. Mr. Mostafa received an inheritance of $7000. He put part of it in a tax shelter paying 9% interest and part in a bank paying 6%. If his annual interest totals $540, how much did he invest at each rate?
Interest. Ms. Jordan invested $4900, part at 6% interest and the rest at 8%. If the yearly interest on each investment is the same, how much interest does she receive at the end of the year?
Interest. If $5000 is invested in a bank that pays 5% interest, how much more must be invested in bonds at 8% to earn 6% on the total investment?
Retail profit. A retailer’s cost for a microwave oven is $480. If he wants to make a profit of 20% of the selling price, at what price should the oven be sold?
Profit from sales. The Beckly Company manufactures shaving sets for $3 each and sells them for $5 each. How many shaving sets must be sold for the company to recover an initial investment of $40,000 and earn an additional $30,000 as profit?
Jogging. Angelina jogs 400 meters in the same time that Harry bicycles 600 meters. If Harry bicycles 60 meters per minute faster than Angelina jogs, find the rate of each.
Overtaking a lead. A car leaves New Orleans traveling 50 kilometers per hour. An hour later a second car leaves New Orleans following the first car, traveling 70 kilometers per hour. How many hours after leaving New Orleans will it take the second car to overtake the first?
Separating planes. Two planes leave an airport traveling in opposite directions. One plane travels at 470 kilometers per hour and the other at 430 kilometers per hour. How long will it take them to be 2250 kilometers apart?
Overtaking a bike. Lucas is of a mile from home, bicycling at 20 miles per hour, when his brother leaves home on his bike to catch up. How fast must Lucas’s brother go to catch Lucas a mile from home?
Overtaking a lead. Karen notices that her husband left for the airport without his briefcase. Her husband drives 40 miles per hour and has a 15-minute head start. If Karen (with the briefcase) drives 60 miles per hour and the airport is 45 miles away, will she catch her husband before he arrives at the airport?
Increasing distances. Two cars start out together from the same place. They travel in opposite directions, with one of them traveling 7 miles per hour faster than the other. After three hours, they are 621 miles apart. How fast is each car traveling?
Travel time. Aya rode her bicycle from her house to a friend’s house, averaging 16 kilometers per hour. It turned dark before she was ready to return home, so her friend drove her home at a rate of 80 kilometers per hour. If Aya’s total traveling time was three hours, how far away did her friend live?
Draining a pool. An old pump can drain a pool in six hours working alone. A new pump can drain the pool in four hours working alone. How long will it take both pumps to empty the pool working together?
Draining a pool. A swimming pool has two drain pipes. One pipe can empty the pool in three hours, and the other pipe can empty the pool in seven hours. If both drains are open, how long will it take the pool to drain?
Filling a blimp. One type of air blower can fill a blimp (or dirigible) in six hours working alone, while a second blower fills the same blimp in nine hours working alone. Working together, how long will it take both blowers to fill the blimp?
Shredding hay. A shredder distributes hay across 5/9 of a field in ten hours working alone. With the addition of a second shredder, the entire field is finished in another three hours. How long would it take the second shredder to do the job working alone?
Sorting letters. A new letter sorting machine works twice as fast as the old one. Both sorters working together complete a job in eight hours. How long would it have taken the new letter sorter to do the job alone?
Plowing a field. A farmer can plow his field by himself in 15 days. If his son helps, they can do it in 6 days. How long would it take his son to plow the field by himself?
Job completion. Suppose an average professor can do a certain job in nine hours and an average student can do the same job in six hours. If three professors and two students work together, how long will the job take?
Butterfat in milk. Two gallons of milk contain 2% butterfat. How many quarts of milk should be drained and replaced by pure butterfat to produce a mixture of 2 gallons of milk containing 5% butterfat? (There are 4 quarts in a gallon.)
Gold alloy. A goldsmith has 120 grams of a gold alloy (a mixture of gold and one or more other metals) containing 75% pure gold. How much pure gold must be added to this alloy to obtain an alloy that is 80% pure gold?
Coffee blends. A coffee wholesaler wants to blend a coffee containing 35% chicory with a coffee containing 15% chicory to produce a 500-kilogram blend of coffee containing 18% chicory. How much of each type is required?
Solder. How many pounds of 60-40 solder (60% tin, 40% lead) must be mixed with 40-60 solder to produce 600 grams of 55-45 solder?
Whiskey proof. How many gallons of 90 proof whiskey (45% alcohol) must be mixed with 70 proof whiskey (35% alcohol) to produce 36 gallons of 85 proof whiskey?
Pharmacy. A pharmacist needs 20% boric acid solution. How much 60% boric acid solution must be added to 7.5 liters of an 8% boric acid solution to get the desired strength?
Mixture of nuts. A mixture of nuts contains almonds worth $9 a pound, cashews worth $11 per pound, and pecans worth $14 per pound. If there are three times as many pounds of almonds as cashews in a 100-pound mixture worth $12.20 a pound, how many pounds of each nut does the mixture contain?
Vending machine coins. A vending machine coin box contains nickels, dimes, and quarters. The coin box contains 3 times as many nickels and 4 times as many quarters as it does dimes. If the box contains a total of $96.25, how many coins of each kind does it contain?
Gumdrops. In a jar of red and green gumdrops, 12 more than half the gumdrops are red and the number of green gumdrops is 19 more than half the number of red gumdrops. How many of each color are in the jar?
Age computation. Eric’s grandfather is 57 years older than Eric. Five years from now, Eric’s grandfather will be 4 times as old as Eric is at that time. How old is Eric’s grandfather?
Real estate investment. A real estate investor bought acreage for $7200. After reselling three-fourths of the acreage at a profit of $30 per acre, she recovered the $7200. How many acres did she sell?
Box construction. An open box is to be constructed from a rectangular sheet of tin 3 meters wide by cutting out a 1-meter square from each corner and folding up the sides. The volume of the box is to be 2 cubic meters. What is the length of the tin rectangle?
Interest rates. Mr. Kaplan invests one sum of money at a certain rate of interest and then half that sum at twice the first rate of interest. This turns out to yield 8% on the total investment. What are the two rates of interest?

Beyond the Basics

You can solve many problems by modeling procedures introduced in this section other than those presented in the examples in the section. Here are some types for you to try.

Suppose you average 75 miles per hour over the first half of a drive from Denver to Las Vegas, but your average speed for the entire trip is 60 miles per hour. What was your average speed for the second half of the drive?
Davinder (D) and Mikhail (M) were 2 miles apart when they began walking toward each other. D walks at a constant rate of 3.7 miles per hour, and M walks at a constant rate of 4.3 miles per hour. When they started, D’s dog, who runs at a constant rate of 6 miles per hour, ran to M and then turned back and ran to D. If the dog continued to run back and forth until D and M met and the dog lost no time turning around, how far did it run?
A mixture contains alcohol and water in the ratio 5:1. After the addition of 5 liters of water, the ratio of alcohol to water becomes 5:2. Find the quantity of alcohol in the original mixture.
An alloy contains zinc and copper in the ratio 5:8, and another alloy contains zinc and copper in the ratio 5:3. Equal amounts of both alloys are melted together. Find the ratio of zinc to copper in the new alloy.
Democritus has lived $\frac{1}{6}$ $\frac{1}{6}$ of his life as a boy, $\frac{1}{8}$ $\frac{1}{8}$ of his life as a youth, and $\frac{1}{2}$ $\frac{1}{2}$ of his life as a man and has spent 15 years as a mature adult. How old is Democritus?
A man and a woman have the same birthday. When he was as old as she is now, the man was twice as old as the woman. When she becomes as old as he is now, the sum of their ages will be 119. How old is the man now?
How many minutes is it before 6 p.m. if, 50 minutes ago, four times this number was the number of minutes past 3 p.m.?
Two pipes A and B can fill a tank in 24 minutes and 32 minutes, respectively. Initially, both pipes are opened simultaneously. After how many minutes should pipe B be turned off so that the tank is full in 18 minutes?
A small plane was scheduled to fly from Atlanta to Washington, D.C. The plane flies at a constant speed of 150 miles per hour. However, the flight was against a head wind of 10 miles per hour. The threat of mechanical failure forced the plane to turn back, and it returned to Atlanta with a tail wind of 10 miles per hour, landing 1.5 hours after it had taken off.
1. How far had the plane traveled before turning back?
2. What was the plane’s average speed?
Three airports A, B, and C are located on an east-west line. B is 705 miles west of A, and C is 652.5 miles west of B. A pilot flew from A to B, had a stopover at B for three hours, and continued to C. The wind was blowing from the east at 15 miles per hour during the first part of the trip, but the wind changed to come from the west at 20 miles per hour during the stopover. The flight time between A and B and between B and C was the same. Find the airspeed of the plane.
Two trains are traveling toward each other on adjacent tracks. One of the trains is 230 feet long and is moving at a speed of 50 miles per hour. The other train is 210 feet long and is traveling at the rate of 60 miles per hour. Find the time interval between the moment the trains first meet until they completely pass each other. (Remember, $1 mile = 5280 feet$ $1 mile = 5280 feet$ .)
Suppose you are driving from point A to point B at x miles per hour and return from B to A at y miles per hour. What is your average speed for the round trip?

Extend the foregoing problem: You travel along an equilateral triangular path ABC from A to B at x miles per hour, from B to C at y miles per hour, and from C to A at z miles per hour. What is your average speed for the round trip?

Critical Thinking/Discussion/Writing

A pawnshop owner sells two watches for $499 each. On one watch, he gained 10%, and on the other watch, he lost 10%. What is his percentage gain or loss?
1. No loss, no gain
2. 10% loss
3. 1% loss
4. 1% gain
The price of gasoline increased by 20% from July to August, while your consumption of gas decreased by 20% from July to August. What is the percentage change in your gas bill from July to August?
1. No change
2. 5% decrease
3. 4% increase
4. 4% decrease

Getting Ready for the Next Section

In Exercises 85–92, simplify each of the expressions.

$\sqrt{8}$ $\sqrt{8}$
$\sqrt{27}$ $\sqrt{27}$
$\sqrt{12}$ $\sqrt{12}$
$\sqrt{45}$ $\sqrt{45}$
$\frac{2 + 3 \sqrt{8}}{2}$ $\frac{2 + 3 \sqrt{8}}{2}$
$\frac{3 + \sqrt{18}}{3}$ $\frac{3 + \sqrt{18}}{3}$
$\frac{15 - \sqrt{75}}{5}$ $\frac{15 - \sqrt{75}}{5}$
$\frac{35 - 14 \sqrt{12}}{7}$ $\frac{35 - 14 \sqrt{12}}{7}$

In Exercises 93–104, factor each of the expressions.

$x^{2} + x$ $x^{2} + x$
$2 x^{2} - 4 x$ $2 x^{2} - 4 x$
$x^{2} - 4$ $x^{2} - 4$
$x^{2} - 25$ $x^{2} - 25$
$x^{2} + 4 x + 4$ $x^{2} + 4 x + 4$
$x^{2} - 6 x + 9$ $x^{2} - 6 x + 9$
$x^{2} - 8 x + 7$ $x^{2} - 8 x + 7$
$x^{2} + 2 x - 15$ $x^{2} + 2 x - 15$
$6 x^{2} - x - 1$ $6 x^{2} - x - 1$
$14 x^{2} + 17 x - 6$ $14 x^{2} + 17 x - 6$
$- 5 x^{2} + 3 x + 2$ $- 5 x^{2} + 3 x + 2$
$- 12 x^{2} + 9 x + 3$ $- 12 x^{2} + 9 x + 3$

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Table of Contents for Section 1.2 Applications of Linear Equations: Modeling

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Table of Contents for
Section 1.2 Applications of Linear Equations: Modeling